# On the chain rule and change of variables of integrals

Originally published at 狗和留美者不得入内. You can comment here or there.

Theorem 1 (Chain rule) Let , , where and are open in , such that are differentiable on their respective domains. Then is also differentiable on , with for all .

Proof: We first assume that there exists a neighborhood of for which . This happens in the case of by inverse function theorem. In that case, by the definition of derivative and its properties, we have In the case of , we have that for all , From this, we easily verifies that , which means that is differentiable at and in the case of , must hold as well. Lemma 1 Let , be differentiable with and . Then, Instead of , one can also use any closed interval of .

Proof: Follows directly from Fundamental Theorem of Calculus. See Theorem 2 (Newton-Leibniz axiom) of . Lemma 1 is a statement of invariance of integral along parameterized smooth paths with the same endpoints.

Theorem 2 (Change of variables or u-substitution in integration) Let be any differentiable function of on , which is continuous on , and be Riemann integrable on intervals in its domain. Then, Proof: Let be an antiderivative of . By the Fundamental Theorem of Calculus, it suffices to show that the left hand side of is equal to , which can be done by applying Lemma 1 accordingly. Theorem 3 (Integration by parts) Let be differentiable functions on and continuous on . Then, Proof: We have Rearranging the above completes the proof. References

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